A033818 Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -2.

3, 5, 9, 14, 22, 34, 53, 83, 131, 208, 332, 532, 855, 1377, 2221, 3586, 5794, 9366, 15145, 24495, 39623, 64100, 103704, 167784, 271467, 439229, 710673, 1149878, 1860526, 3010378, 4870877, 7881227, 12752075, 20633272, 33385316, 54018556, 87403839, 141422361, 228826165, 370248490

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Formula

a(n) = L(1)*(F(n+1) - 1) + L(0)*F(n) - L(-1)*n, F(n): Fibonacci (A000045), L(n): Lucas (A000032) with L(-n) = (-1)^n*L(n).

G.f.: x*(3-4*x)/((1-x-x^2)*(1-x)^2).

a(n) = Lucas(n+1) + n - 1. - G. C. Greubel, Jun 01 2019

Mathematica

LinearRecurrence[{3, -2, -1, 1}, {3, 5, 9, 14}, 50] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2011, modified by G. C. Greubel, Jun 01 2019 *)
Table[LucasL[n+1] +n-1, {n, 1, 50}] (* G. C. Greubel, Jun 01 2019 *)

Prog

(PARI) {a(n) = fibonacci(n+2) + fibonacci(n) + n-1}; \\ G. C. Greubel, Jun 01 2019
(Magma) [Lucas(n+1) +n-1: n in [1..50]]; // G. C. Greubel, Jun 01 2019
(Sage) [lucas_number2(n+1, 1, -1) +n-1 for n in (1..50)] # G. C. Greubel, Jun 01 2019
(GAP) List([1..50], n-> Lucas(1, -1, n+1)[2] +n-1) # G. C. Greubel, Jun 01 2019

Crossrefs

Cf. A000032, A000045, A023537, A023548, A033811.

Sequence in context: A267047 A032801 A332641 * A320598 A227567 A120452

Adjacent sequences: A033815 A033816 A033817 * A033819 A033820 A033821

Keyword

easy,nonn

Author

Wolfdieter Lang

Extensions

Terms a(31) onward added by G. C. Greubel, Jun 01 2019

Status

approved